SLINKY WAVES
INTRODUCTION
In this experiment you will be investigating the propagation of waves in two different types of systems.
These systems are designed to make the wave motion slow enough to allow you to view directly a
number of features of wave motion. We hope that by the end of the experiment you will have developed
a number of intuitions about how waves travel, that you will have learned some new properties of wave
motion which, although common occurrences are not talked about in introductory physics courses, and
that you will have some level of quantitative understanding of wave propagation.
Much of this investigation is qualitative. In introducing a number of phenomena we try to give a
conceptual explanation. In this process we shall be quoting formulae and relations that you should be
able to understand and the results of which you should be able to use. However, many of these formulae
involve theory beyond the reach of students in an introductory physics course, so we do not expect that
you will be able to follow the derivations, which we shall thus not detail.
THEORY AND BACKGROUND TO THE EXPERIMENT
You shall be investigating wave propagation in two types of media: dispersive and non-dispersive. Nondispersive propagation is what is generally studied in high school, is simpler to deal with, but is probably
less common in nature. Non-dispersively propagated waves move at a constant speed, independent of
the frequency of the wave disturbance. An example of this is wave motion on a stretched string. On the
other hand dispersive propagation involves a variation of speed of the wave with frequency. Examples of
this are waves in water and light waves moving through glass.
OUR NON-DISPERSIVE SYSTEM
You shall be investigating non-dispersive propagation using the Shive wave machine. This consists of a
series of rods having considerable moment of inertia, with adjacent rods coupled by a weak torsion
(spring) wire. The wire provides a restoring force proportional to the difference of angular displacement
between adjacent rods, and thus a displacement of one rod produces a torque on its neighbours which
accordingly accelerate. A wave motion results because the displacement of one rod then produces a
delayed displacement of its adjacent rods, the delay depending on the moment of inertia of the rod and
the torsion constant of the joining spring wire. (For a mathematical expression of this statement, refer to
one of the references listed at the end of this guide sheet.)
If we let y be the angular displacement of the rod and x be the position along the string of rods, then for
pure sine waves driving the whole system (i.e., for motions with pure single frequencies) the wave is
described by:
yr = Ar cos(ωt − kx + ϕ r )
(1a)
yl = Al cos(ωt + kx + ϕ l )
(1b)
y = y r + yl
(1c)
where yr and yl represent waves moving to the right and left respectively, and y represents the general
case. We have the angular frequency
k=
2π
λ
ω = 2πf , where f is the frequency of oscillation as seen from one position in space.
is the wave number, where λ is the wavelength as viewed at one moment in time.
For our non-dispersive (Shive) wave machine, wave speed is constant.
c = fλ =
ω
k
1
Thus there exists a constant co such that:
ω =c 0 k
(2)
c0 is then the speed of the wave, independent of ω.
OUR DISPERSIVE SYSTEM
You shall be investigating dispersive propagation using the suspended slinky. This consists of a coil
spring which can viewed as a series of coil loops (which have mass) which are coupled together by the
spring constants of the coils themselves. Thus when one loop is displaced by a distance y along the axis
of the spring, the springiness of the turns of the slinky produces a force on its neighbours which
accordingly accelerate. A wave motion results because the displacement of one coil loop then produces
a delayed displacement of its adjacent loops, the delay depending on the mass of the loop and the spring
constant of the joining components between the loops. So far the situation described is totally analogous
to the description of the Shive machine above, with the longitudinal translational motion here being the
analogue of the Shive’s rotational motion. In this analogue, x (in equations (1a), (1b), (1c)) represents a
distance along the slinky, and y represents the displacement of a coil (at position x) from its normal
resting position. Thus the equations quoted for the Shive system should also apply here, and the slinky
would be non-dispersive. This would be correct if the slinky were resting on a friction-free skating rink.
However, our slinky is suspended by strings of a fixed length, so that each coil loop thinks itself to be a
pendulum of length equal to the string length and mass equal to the loop mass. Thus, two sets of forces
act on each loop, one as described above which produces wave motion, and one from gravity which
provides a restoring force to make the loop swing back to its centre position. We shall, in this experiment,
find that the pendulum component makes the slinky waves dispersive.
It can be shown that, for this suspended slinky system, ω and k are related in a slightly more complicated
form than that of equation (2). The relationship is:
ω = ω02 + c02 k 2
(3a)
ω 2 − ω02
k=
(3b)
c02
Here ωo is the angular frequency for a pendulum of length of the slinky suspension string system and c0
is, as before, the constant wave speed of a non dispersive system, in this case consisting of the slinky
resting on a frictionless surface.
Note that since the slinky can be driven by a motor at any frequency, ω can assume a variety of values.
Equation (3b) leads us to investigate three ranges of values of ω.
Range 1: ω > ωo
In this case k2 > 0 , so k is a real number. Equations (1a), (1b), (1c) fully describe the wave motion of the
slinky under these conditions, and the speed of phase propagation of these waves is
c=
ω
k
=
ω c0
ω −ω
2
(4)
2
0
This case, at higher frequencies, produces wave motion because the high acceleration of the coil loops
makes the effect of the inertia of each loop predominate over the pendulum effect due to the strings.
In your experiment with the slinky, one end of the slinky is held fixed and the other may be driven
sinusoidally. For convenience take the fixed end to be x ≡ 0; the fact that it is fixed implies y = 0 at x = 0
for all t; hence, from (1a), (1b), (1c) you can derive, with proper choice of the origin of time, the solution:
y = y0 sin ω t sin kx
(5)
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where y0 is a constant. The wave in this case is a standing wave rather than a propagating wave – it is a
superposition of two waves of equal amplitudes propagating in opposite directions: “nodes” of zero
displacement are separated by distances π/k , half the “wavelength” 2π/k of the propagating waves.
The amplitude y0 is determined by the driving motor at the end of the slinky where x = L; if the motor
enforces a sine wave motion with amplitude yd there (i.e., y = yd sinωt at x = L), then from (5)
yd sin ω t = y0 sin ω t sin kL
yd
sin kL
And so: y 0 =
(6)
(7)
Evidently y0 will tend toward infinity (being limited only by dissipation and nonlinear effects) for any k such
that: k = k n =
nπ
, n = 1,2,3.......
L
(7’)
This represents the “resonant” behaviour; the different natural modes of oscillation represented by (7),
called the normal modes or the “eigenmodes” – exhibit one loop, two loops, three loops, etc., between the
two ends; they may be thought of as the fundamental (or first harmonic), first overtone (second
harmonic), second overtone (third harmonic), etc. Note, however, that the harmonics are characterized
by k being some integral multiple of the fundamental value π/L; the associated ω, from (3a) does not
progress in steps of integral multiples, but instead
n 2 c02π 2
ω = ω +c k = ω +
L2
2
0
2 2
0 n
2
0
(8)
(This is different from the non-dispersive Shive machine, or the violin string in which:
ωn =
nc0π
)
L
It might be pointed out that the human ear finds pleasure in combinations of notes whose frequencies are
integer multiples of a fundamental frequency; octaves have a frequency ratio of 2:1, fifths have a 3:2
frequency ratio, major thirds a 5:4 ratio, etc. Thus non-dispersive violin strings can produce pleasant
sounds.
Range 2: ω < ωo
In this case k2 < 0 , so k is an imaginary number. This would imply that no real wave could be
propagated. Thus, solutions (1a), (1b) and (1c) do not work. The solutions for this case are:
Where: k =
2
y r = Ar cos(ω t + ϕ r ) e − kx
(9a)
yl = Al cos(ω t + ϕ l ) e + kx
(9b)
y = yl + y r
(9c)
ω02 − ω 2
c02
(10)
ω = 2πf , where f is the frequency of oscillation as seen from one position in space
k=
1
1
, where x0 is the exponential decay distance to of the amplitude.
e
x0
This case, at lower frequencies, produces no wave motion because the pendulum effect due to the
restoring force of gravity on the coil loops hung from the strings predominates over inertial effects. Thus
each coil sees its adjacent coil pulling it to one side, while gravity tries to pull it back to its central position.
Here the whole slinky moves in unison, the amplitude of motion decreasing exponentially with distance
along the slinky. This solution of equations (9a), (9b), (9c) produces two “waves” which are called
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“evanescent”; they are stationary, rather than propagating in the x direction, one growing and the other
decaying exponentially with increase of x, the “e-folding length” being k -1.
With an infinitely long sinusoidal driven spring, the relevant solution would of course be the one that
decayed exponentially away from the driven point (yr being 0 on one side, yl being 0 on the other side). In
the present experiment, with the fixed end taken as x ≡ 0 so that y = 0 at x = 0, from (9a), (9b) and (9c)
and with the proper choice of the origin of time we have the solution:
(11)
+ kx
− kx
y = y0 sin ω t (e
−e
)
As before, the amplitude y0 is determined by the driving motor at x = L where it enforces a sine wave
motion with amplitude yd (i.e., y = yd sinωt at x = L), so
y0 =
e
+ kL
yD
− e − kL
(12)
Note that there are no resonances available here; y is necessarily a monotonic function of x, so there are
no nodes between the end points; the wave is a standing or stationary wave, but one without the familiar
node-and-loop pattern.
The limiting case of ω → 0 (approximated by taking one end of the suspended slinky and holding it with a
fixed displacement) shows the exponential “decay” of the displacement along the slinky. We have:
k=
ω02 − ω 2
c0
→
ω0
c0
In this case, if the pendulum restoring force is much weaker than the spring restoring force (as it is in our
slinky), so that
c0
ω0
is very small compared to the apparatus dimensions, then for reasonable distances
away from the fixed end, kL ≥ kx >> 1 so that
y ≅ y0 e kx sin ω t and the displacement of the coils is
exponential.
Range 3: ω = ωo
This is the limiting case between the previous two ranges, with k1,2 = 0
For our already stated boundary conditions of y = 0 at x = 0, and y = yd sinωt at x = L gives the solution
y = yD
x
sin ω t
L
(13)
with the amplitude diminishing linearly from yd at the driven end to 0 at x = 0.
THE APPARATUS
As has already been pointed out, you have two apparati to work with: a slinky and a Shive machine.
The slinky is fixed at one end and has a high precision sinusoidal drive at the other. You should inspect
the drive to assure yourself that the motion is truly sinusoidal. The drive motor speed is continuously
variable and is adjusted from a control box. The drive has an electronic timer associated with it which
measures the time between successive oscillations. The timer has an accuracy of ±0.001 s. It should be
noted that the slinky system has quite a high “Q” value, which means that little energy gets lost from the
slinky spring in each oscillation. A result of this is that modes of motion that get started take a long time
to decay away and it takes a long time for new modes to start up. Thus in varying the frequency (speed)
of the drive it is important to make the changes slowly or at least to wait a long time after a sudden
change is made to observe the steady state result of the change.
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You will note the three sets of “combs” on the apparatus. These may be raised or lowered to change the
effective string length and thus change ωo. A 3 mm diameter steel rod is provided to facilitate the
lowering or raising of the combs; the rod gets inserted in the hole near the edge of the comb.
The Shive machine, as we have it, is a lower precision device. However, it does provide a means of
observing the (less interesting) case of non-dispersive wave motion. There are two versions of the
sinusoidal drive for the Shive machine. One is similar to the slinky’s drive. The other is a simpler motor
which can be attached to the end rods of the apparatus. This motor has a cruder speed control, gives
motion that is not so precisely sinusoidal, and requires you to time oscillations with a stopwatch (you will
find that you will develop considerable skill at high speed timing with this version of the apparatus). A
simple clamp is provided so that you can fix the undriven end, if you wish.
THE EXPERIMENT
The following suggests a number of things you can do to check out the two systems. Investigate the
wave characteristics according to your interest.
1. For both the Shive wave machine and the suspended slinky start a “pulse” off at one end and time
it over the round trip to the other end and back. This is done by giving an end rod of the Shive
apparatus one sudden up-down shake or by giving one lateral shake to a coil near one end of the
slinky. Note the continued sharpness of the Shive pulse while for the slinky note that the pulse that
returns is not as sharply defined as the one you send; this is to a lesser part the result of dissipation
(whose effects have not been discussed) but principally is non-dissipative “dispersion” caused by the
fact that c in equation (4) is not precisely constant for the range of ω’s of your pulse. Find,
approximately, c0 for both apparati.
2. Deduce ωo for the suspended slinky by direct measurement of the vertical length of the strings, with
filtering “combs” out of the way.
3. Î Python requirement
Drive the slinky at a variety of ω’s in the “propagating” range (ω > ωo); measure distance between
and
adjacent nodes, and so determine corresponding k’s. Plot ω2 against k2 and deduce
from a linear fit according to equation (3b); compare with values inferred in parts (1) and (2). After
each measurement, turn the driving motor off suddenly; note that, in general, the oscillations persist
in a haphazard fashion, reflection back and forth and gradually dissipating, but clearly not continuing
as simple oscillations (at a given point) at the original ω. This haphazardness does not exist for
normal modes as you will find in part (5).
4. Î Python requirement1
For the Shive apparatus, drive the machine at a variety of ω’s as in part (3), using the drive motor at
one end and clamping the last two rods at the other end to provide y = 0 at x = 0. Equation (5) applies
just as well to describe standing waves formed in this manner on the Shive machine; measure the
distances between adjacent nodes and from this deduce the values of k. Do the values of
non-dissipative propagation?
suggest
5. For the slinky, find the family of resonant ω’s within the range of ω’s provided by your driving
motor (but do not destroy the suspension in the process). Are they integral multiples of some
fundamental angular frequencies, or is the more general relation of equation (8) required to describe
them? For each resonant mode, turn off the motor abruptly. Note that the mode remains in situ,
decaying gradually and not haphazardly.
6. For the Shive apparatus, find the family of resonant ω’s, as in part (5), using the drive motor at
one end and clamping the last two rods at the other end to provide y = 0 and x = 0.
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7. Î Python requirement1
Drive the slinky at an ω << ωo. Does the amplitude fall off exponentially with distance from the driven
end? Or are both exponentials in equation (11) required to describe it? Collect some data and
perform a nonlinear fit, trying one or a sum of two exponentials as your model. Can you pick in
advance an ω that will lead to kL >> 1 and hence a simple exponential decay near the driven end?
Try the extreme of ω → 0 by merely displacing and holding stationary one end of the slinky and noting
the displacements (y’s) of the coils down the apparatus.
8. Î Python requirement1
Having inferred ωo from part (2) or, probably better, from part (3), drive the slinky at ω = ω0. Collect
some data and perform a linear fit to test how accurately the linear decay law of equation (13) is
obeyed. Discrepancies will indicate imperfections of suspension, of uniformity in the slinky, or
perhaps of dissipative effects. Can you distinguish between them?
9. When a filtering comb is lowered, meshed with the supporting strings, it effectively shortens the ℓ
and so raises the ω0 of that portion of the slinky. Let ωℓ be the ω0 for the original long strings and ωs
be the ω0 for the “short” strings, with ωℓ < ωs. Then there exist ω’s that satisfy ωℓ < ω < ωs, such that
corresponding waves may propagate through the long-string (“ℓ”) section of the slinky but are
prevented from propagating through the short-strong(“s”) section: the “s” section then acts as a filter
for the removal of propagating waves. As you will see in the next exercise, wave energy can,
however, “tunnel” through the “s” section, particularly if ksLs is not >> 1 (where ks and Ls are the k and
length of the “s” sections).
10. Try lowering combs number 2 and number 3, away from the driven end. Now you can establish a
resonance belonging essentially to section number 1, nearest the drive; since sections number 2 and
number 3 will not support propagation, they behave somewhat as if they moved the fixed point x = 0
toward the driven end (in the sense that they produce only small amplitudes of oscillation), and the
loop(s) is (are) confined primarily to section number 1.
11. Now lower combs number 1 (nearest to the driven end) and number 2, raising comb number 3.
Drive at the same frequency. Does section number 3 exhibit resonance? Can the energy from the
driving motor indeed “tunnel” through sections number 1 and number 2, to establish a resonant
response in section number 3? Does section number 3 exhibit larger amplitudes than sections
number 1 and number 2, despite being farther from the driven end?
12. With comb number 1 raised and with section number 1 resonating, the combs in the other two
sections being lowered (as in part (10)), determine the resonance wave period. Then turn off the
driving motor abruptly and simultaneously raise the comb in section number 3 so that section too
could resonate if only it got energy at the appropriate frequency – namely, the frequency at which the
other section is oscillating. Watch (in fascination) as the energy “tunnels” through the intervening
section number 2, and then “tunnels” back again, in a fashion reminiscent of loosely coupled pendula,
though now more dissipative. (Section number 2 provides the coupling, and the fact that it is a nonpropagating region is what makes the coupling “loose”.) Can you determine the beat period of
tunnelling?
13 Section (12) suggests that the apparently resonant mode with which you started has become a
superposition of two resonant modes, beating together, with the lifting of the comb number 3. Find
them (at frequencies slightly higher and lower than the original frequency). Note that one is a
“symmetric” mode (y(x) = y(-x) when x is measured from the centre) and the other is an
“antisymmetric” mode (y(x) = -y(-x) when x is measured from the centre). Compare the difference of
resonance frequencies with the “beat” frequency of tunnelling, if you are able to determine the latter
reliably.
For those who did not intend to do all three weights of this experiment, I would suggest that parts (1) and (2) are fast and useful, and
parts (5), (7), (12), (13) are probably most interesting.
REFERENCES
A.P. French, Vibrations and Waves, Norton 1971, Chapters 5, 6, 7, 8
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Python requirements are ONLY for PHY224/324H1students
(Revised by Sergei Sagatov and Andrew Martin In 2009. Older versions: ch – 1980, jv – 1988)
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